Central limit theorem for Fourier transforms of stationary processes

Peligrad, Magda; Wu, Wei Biao

doi:10.1214/10-aop530

Cited by 55 publications

(91 citation statements)

References 30 publications

(33 reference statements)

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“…Theorem 11 is very general and it allows nonlinear, non-strong mixing and/or even long-memory processes. It follows from Theorem 1 in Peligrad and Wu (2010). Proposition 2 concerns a fixed frequency ϑ ∈ (0, 2π) and it is established in Wu (2005).…”

Section: Definition 4 (Periodogram) Let ı =mentioning

confidence: 82%

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Asymptotic theory for stationary processes

2011

Statistics and Its Interface

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105

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We present a systematic asymptotic theory for statistics of stationary time series. In particular, we consider properties of sample means, sample covariance functions, covariance matrix estimates, periodograms, spectral density estimates, U -statistics, kernel density and regression estimates of linear and nonlinear processes. The asymptotic theory is built upon physical and predictive dependence measures, a new measure of dependence which is based on nonlinear system theory. Our dependence measures are particularly useful for dealing with complicated statistics of time series such as eigenvalues of sample covariance matrices and maximum deviations of nonparametric curve estimates.

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Section: Definition 4 (Periodogram) Let ı =mentioning

confidence: 82%

“…Theorem 1 in Peligrad and Wu (2010) asserts that, for a regular process, its spectral density function exists almost surely over φ ∈ [0, 2π] with respect to the Lebesgue measure. If…”

Section: Definition 4 (Periodogram) Let ı =mentioning

confidence: 99%

Asymptotic theory for stationary processes

2011

Statistics and Its Interface

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105

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“…That is, the quantum state of the beam is that of a collimated, polarized, filtered thermal beam (16).…”

Section: Resultsmentioning

confidence: 99%

“…However, Gaussian statistics in the k-space are actually generic in the following sense [16]: if u(t) is a stationary ergodic process, then the limit as L → ∞ ofũ L (k) is a complex Gaussian random variable of uniformly random phase, whose statistics are thus completely defined by its second moment…”

Section: Resultsmentioning

confidence: 99%

How many principles does it take to change a light bulb…into a laser?

Wiseman¹

2016

Phys. Scr.

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Abstract. Quantum optics did not, and could not, flourish without the laser. The present paper is not about the principles of laser construction, still less a history of how the laser was invented. Rather, it addresses the question: what are the fundamental features that distinguish laser light from thermal light? The obvious answer, "laser light is coherent", is, I argue, so vague that it must be put aside at the start, albeit to revisit later. A more specific, quantum theoretic, version, "laser light is in a coherent state", is simply wrong in this context: both laser light and thermal light can equally well be described by coherent states, with amplitudes that vary stochastically in space. Instead, my answer to the titular question is that four principles are needed: high directionality, monochromaticity, high brightness, and stable intensity. Combining the first three of these principles suffices to show, in a quantitative way -involving, indeed, very large dimensionless quantities (up to ∼ 10 51 ) -that a laser must be constructed very differently from a light bulb. This quantitative analysis is quite simple, and is easily relatable to "coherence", yet is not to be found in any text-books on quantum optics to my knowledge. The fourth principle is the most subtle and, perhaps surprisingly, is the only one related to coherent states in the quantum optics sense: it implies that the description in terms of coherent states is the only simple description of a laser beam. Interestingly, this leads to the (not, as it turns out, entirely new) prediction that narrowly filtered laser beams are indistinguishable from similarly filtered thermal beams. I hope that other educators find this material useful; it may contain surprises even for researchers who have been in the field longer than I have.2

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“…For stationary processes, joint Gaussianity of frequency coefficients has been established via a central limit type argument [135], [184], [185]. The proofs of these works are not reproduced here, but the intuition behind them is that the frequency coefficients are obtained by summing a relatively large number of random variables multiplied by orthogonal complex exponential functions, which leads to a central limit theorem for wide-sense stationary processes.…”

Section: Joint Gaussianity Of Frequency Coefficientsmentioning

confidence: 99%

Graphical model driven methods in adaptive system identification

Yellepeddi¹

2016

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Identifying and tracking an unknown linear system from observations of its inputs and outputs is a problem at the heart of many different applications. Due to the complexity and rapid variability of modern systems, there is extensive interest in solving the problem with as little data and computation as possible.This thesis introduces the novel approach of reducing problem dimension by exploiting statistical structure on the input. By modeling the input to the system of interest as a graph-structured random process, it is shown that a large parameter identification problem can be reduced into several smaller pieces, making the overall problem considerably simpler.Algorithms that can leverage this property in order to either improve the performance or reduce the computational complexity of the estimation problem are developed. The first of these, termed the graphical expectation-maximization least squares (GEM-LS) algorithm, can utilize the reduced dimensional problems induced by the structure to improve the accuracy of the system identification problem in the low sample regime over conventional methods for linear learning with limited data, including regularized least squares methods.Next, a relaxation of the GEM-LS algorithm termed the relaxed approximate graph structured least squares (RAGS-LS) algorithm is obtained that exploits structure to perform highly efficient estimation. The RAGS-LS algorithm is then recast into a recursive framework termed the relaxed approximate graph structured recursive least squares (RAGS-RLS) algorithm, which can be used to track time-varying linear systems with low complexity while achieving tracking performance comparable to much more computationally intensive methods.The performance of the algorithms developed in the thesis in applications such as channel identification, echo cancellation and adaptive equalization demonstrate that the gains admitted by the graph framework are realizable in practice. The methods have wide applicability, and in particular show promise as the estimation and adaptation algorithms for a new breed of fast, accurate underwater acoustic modems.The contributions of the thesis illustrate the power of graphical model structure in simplifying difficult learning problems, even when the target system is not directly structured.Thesis Supervisor: James C. Preisig Title: Scientist Emeritus, Woods Hole Oceanographic Institution AcknowledgmentsPiglet noticed that even though he had a Very Small Heart, it could hold a rather large amount of Gratitude. Winnie the PoohA.A.MilneMy years at MIT have been about so much more than a degree. In the past six years you have introduced to more worlds than I ever knew existed. As this long and wonderful journey ends, I find that my heart can, too, hold so much gratitude, for everything that you have given.Of course, I haven't said who "you" is, so here goes.It is impossible to write any list of acknowledgements without having at the very top my thesis advisor Dr. James Preisig, who has truly been a wide-angle lens f...

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Central limit theorem for Fourier transforms of stationary processes

Cited by 55 publications

References 30 publications

Asymptotic theory for stationary processes

Asymptotic theory for stationary processes

How many principles does it take to change a light bulb…into a laser?

Graphical model driven methods in adaptive system identification

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